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Properties

Label 2-3200-40.13-c0-0-5
Degree $2$
Conductor $3200$
Sign $0.850 - 0.525i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·9-s + (1 − i)13-s + (−1 + i)17-s + 2·29-s + (1 + i)37-s + i·49-s + (−1 + i)53-s − 2i·61-s + (1 + i)73-s − 81-s + (1 − i)97-s + (1 + i)113-s + (1 + i)117-s + ⋯
L(s)  = 1  + i·9-s + (1 − i)13-s + (−1 + i)17-s + 2·29-s + (1 + i)37-s + i·49-s + (−1 + i)53-s − 2i·61-s + (1 + i)73-s − 81-s + (1 − i)97-s + (1 + i)113-s + (1 + i)117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281287437\)
\(L(\frac12)\) \(\approx\) \(1.281287437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.630053899975606689695176076637, −8.253870150607420174793273787104, −7.61763891168823077212505296624, −6.45098444084630268463399899097, −6.06862884331546124309552301122, −4.98006158610815547455180929820, −4.38038051118067235606833245271, −3.29244737107063561412180363178, −2.41924396642072560160029415616, −1.25243281903463886346800228200, 0.914987420435682091305445612252, 2.22088826751870483828544639037, 3.24224194484881919221777398388, 4.13953403024065277636310202923, 4.78761345212855034210472940620, 5.95113047459660457500579286515, 6.58140971038435988410035407164, 7.06396529055362152126001760228, 8.186138585564438907729875065059, 8.898343517022490814201378567211

Graph of the $Z$-function along the critical line